The series' central object is not G but its iterate: g⁰ carried to g³³ under repeated application. In a strict setting G^[33] is unambiguous. In a weak one, every application introduces coherence data, and thirty-three applications introduce a great deal of it.
The number of bracketings of a 33-fold composite is the 32nd Catalan number — on the order of
1017. Mac Lane's coherence theorem is precisely the statement that this does not matter:
all of them are canonically isomorphic, provided the pentagon holds. So chapter 4 is not a technical
preliminary to this chapter. It is the entire licence for writing G^[33] at all.
Coherence says the bracketings agree. It does not say the composite has a normal form, and it does not say anything about what is preserved across the 33 steps. The series already has a candidate for that: the transverse Floquet multiplier λ⊥ = e−4π, established elsewhere in the corpus as a contraction rate under iteration. A 2-categorical reading would ask whether that multiplier is an invariant of the composite or an artefact of one bracketing.
Show that the 33-fold composite admits a canonical representative, and
that the quantity the series attaches to it — the contraction rate, the attractor claim, the
bound ∃ m ≤ 33 appearing in the AXLE theorems — is independent of bracketing. If it
is not, the bound is a statement about a bracketing and not about the system.
A proof, or a counterexample, that the series' 33-step
quantities are bracketing-independent. Until then ∃ m ≤ 33 is under-specified in the
same way an unbracketed composite is.